A unified equilibrium framework of new shared mobility systems

Transportation Research Part B 129 (2019) 50–78

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

A unified equilibrium framework of new shared mobility systems Xuan Di a,∗, Xuegang Jeff Ban b,c a

Department of Civil Engineering and Engineering Mechanics, Columbia University, USA Department of Civil and Environmental Engineering, University of Washington, Seattle, USA c College of Transport and Communications, Shanghai Maritime University, China b

a r t i c l e

i n f o

Article history: Received 31 October 2018 Revised 2 July 2019 Accepted 2 September 2019

Keywords: Equilibrium Ridesharing Transportation network companies (TNC) Super extended network

a b s t r a c t Modeling congestion effects arising from multiple travel modes, shared mobility modes in particular, is non-trivial because of the complex interactions among diverse agents and distinct traffic flow compositions. This research aims to provide a theoretical framework of generic traffic network equilibria to unify these services and hopefully become a step stone to modeling shared mobility services in congested road network. In the proposed framework, we mainly focus on three modes: driving solo, ridesharing, and e-hailing service. The four types of traffic flows are: personal vehicle drivers, e-hailing drivers, ridesharing riders, and e-hailing passengers. The first two flows contribute to traffic congestion while the latter two do not. To capture their interactions, a super extended network is created with four copied networks each of whom represents one type of traffic flow. The equilibrium of new mobility systems can then be reformulated as a quasi-variational inequality and solution existence is discussed. The numerical results are tested in both Braess network and Sioux Falls network to illustrate the impact of different parameters on equilibrium outcomes, including modal cost, system travel time and deadhead miles. The results of this model will help assist transportation planners in making policy and regulation decisions regarding shared mobility services. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Shared mobility (such as ridesharing and e-haling) has been rapidly growing in the past few years and become more and more popular for their convenient (or on-demand) and relatively cheap service (Furuhata et al., 2013; Shaheen et al., 2016). However, it remains unclear how the introduction of shared mobility into the existing transportation system affects travelers’ mode choice behavior and network-wide congestion. Such uncertainty poses challenges for urban planners and policy-makers to evaluate the performance of these emerging mobility service and to propose effective countermeasures to regulate these services. Unfortunately neither the conventional transportation planning models nor management tools take the wide scale adoption of these new services into consideration, which may produce inaccurate traffic prediction or forecast. This paper aims to develop a modeling framework which accommodates these new services (ridesharing, and ehailing service in particular) and analyze their impacts on traffic network congestion in the new landscape where multiple travel modes co-exist. ∗

Corresponding author. E-mail address: [email protected] (X. Di).

https://doi.org/10.1016/j.trb.2019.09.002 0191-2615/© 2019 Elsevier Ltd. All rights reserved.

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Because of deadhead miles incurred by unoccupied vehicles searching for customers, it has long been debated that the transportation network companies (TNC) like Uber potentially aggravate traffic congestion (Flegenheimer, 2015). This debate raised a critical research question as to whether the introduction of new ride-hailing services makes transportation systems worse. TNCs has also been blamed to captivate customers from existing travel modes (e.g., the traditional taxi industry), which reveals another key question as to the relationship between e-hailing services and the existing travel modes. Exploring these relations helps assist transportation planners in making policy and regulation decisions regarding shared mobility services. The existing data could fail to provide a comprehensive answer to the aforementioned questions (NYC.gov, 2016; Li et al., 2016), partly because these new services are still growing and their market has not gained a dominant number of users over some existing travel choices. A projection into the future based on transient behavioral data may generate biased outcomes. This research thus aims to develop a more generic network equilibrium framework which accounts for the interactions among different types of travel modes (ride-hailing services in particular) and anticipates to offer a scientific and comprehensive solution to the above focal research questions. 1.1. Literature review The traffic assignment problem (TAP) models the interaction between transportation infrastructure (i.e., hard supply) and travelers (i.e., demand) in order to predict traffic congestion. The conventional TAP, assuming one unit of traveler is served by one unit of vehicular flow (i.e., single-occupancy vehicle (SOV)), is not suitable to model the nascent shared mobility services such as ridesharing and TNC services. These services separate person flow (which accounts for travel demands and may not contribute to congestion but can induce vehicular flow) from vehicular flow (i.e., soft supply, which serves person flow and accounts for traffic congestion) and thus complicate the modeling of TAP. Ridesharing (RS) (also called carpooling, not-for-profit ridesharing, or organized ridesharing (Furuhata et al., 2013)) has gained growing interests in recent years due to its capability of reducing travel cost, relieving traffic congestion, and lessening energy consumption. Modeling RS as a traffic network equilibrium dated back to the carpooling equilibrium developed by Daganzo (1981). The carpooling equilibrium is one specific case of the multimodal-equilibrium where each vehicle type or travel mode has different travel cost structure and thus contributes differently to the overall congestion. As an extension, Song et al. (2015) formulated the route choice part of multimodal equilibrium as a variational inequality and mode choice as a logit model. Dynamic traffic assignment (Abdelghany et al., 20 0 0; Murray et al., 2001) and stochastic dynamic user optimal (He et al., 2003) were also proposed. These models, however, did not explicitly model passenger flow and thus ignored passengers’ mode choice behavior. Xu et al. (2015) proposed a more general complementarity formulation of ridesharing user equilibrium (RUE) using an extended network structure. To accommodate three types of travelers (i.e., solo drivers, ridesharing drivers, and passengers), each link in the network has three copies where each copy represents one traveler class; each node has two copies: solo drivers and ridesharing drivers share one set of nodes while passengers use the other set of nodes. At equilibrium, all travelers share the same generalized travel cost on each path and thus nobody can improve her travel cost by unilaterally switching routes or roles. Di et al. (2017), Li et al. (2019), Di et al. (2018) further studied the impacts of building HOT lanes and subsidy on RUE and the system performance. Self-organizing RS service can be uncertain given drivers are not bound to provide service for profit. Taxi or e-hailing drivers, on the other hand, are driven by profit and thus are more committed to fulfilling others’ travel requests. Note that e-hailing service is a broader term which can be provided by a TNC vehicle or an app-based taxi. There are a large body of studies on modeling supply-demand equilibria of a taxi market with both traditional and e-hailing taxis (Yang and Wong, 1998; Wong et al., 2008; Yang et al., 2010a; He and Shen, 2015; Qian and Ukkusuri, 2017). In a taxi market, there exist multiple interactive entities exhibiting different equilibrium behaviors: passengers can be treated as leaders who choose service providers in order to minimize travel disutility, and traditional taxi or app-based taxi are followers responding to passengers’ choices for profit maximization at a cost of matching and searching friction. To capture the interplay of these behaviors, a Stackelberge game (He and Shen, 2015) or multi-leader-follower game (Qian and Ukkusuri, 2017) was formulated. However, vehicles’ route choice behavior and network level congestion were not modeled. Wong et al. (2001a) extended taxi movements to congested road networks with elastic demand. Provided that taxi is the only travel option, a long waiting time resulting from traffic congestion simply oppresses some passengers’ travel needs. Unlike taxi companies, TNCs have no absolute control over drivers nor vehicle fleets. Drivers behave strategically about when and where to work. A two-sided market forms wherein the TNC platform leverages pricing to balance supply and demand. There exists a few studies which captures the strategic decisions among drivers, passengers, and the platform using a queuing-theoretic economic model (Banerjee et al., 2016). Within the queuing network, travel time is random and roads are treated as infinite servers, i.e., travel time may be time-dependent but flow-independent. To simplify the model structure, drivers waiting and travel behavior is modeled while passengers simply arrive at certain locations in distribution. Computation of such equilibrium is quite involved and a close form is obtained only when Markovian properties are assumed. Although the aforementioned studies modeled the interactions among entities involved in a transportation system, the mainstream research is focused on a single-mode flow (i.e., either conventional taxi or e-hailing) without considering its interactions with other traffic flows. Accordingly the movement of these service vehicles in a network experiences flowindependent travel time. This assumption holds when neither taxi nor TNCs constitutes a majority of traffic flow. In major cities like NYC or HK, however, taxi is one indispensable component of traffic flow and thus contributes to traffic congestion to a large extent. Ignoring its impact on traffic congestion might lead to inaccurate prediction of traffic distribution or biased

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assessment of these new services. On the other hand, traffic congestion cost plays an important role when travelers make decisions about their mode and route choice, especially during peak hours. With the growing popularity of personal mobility services, their impacts on traffic congestion cannot be ignored. Unfortunately, modeling congestion effects arising from multiple travel modes, shared mobility modes in particular, is non-trivial because of the complex interactions among diverse agents and distinct traffic flow compositions. To the best of our knowledge, there exists only a few studies which integrated the behaviors of different transportation entities and their interactions into TAP. Yang et al. (2010a, 2014) added a “virtual” non-taxi mode with a constant utility (e.g., public transit) to accommodate bi-modal selection. Wong et al. (2008) used hierarchical logit to model mode split between taxis and normal traffic, and among different taxi vehicle types. A recent example is (Ban et al., 2018) to model the interactions of solo driving, e-hailing taxis, and TNC services. Although these studies considered the interactions of taxi or TNC vehicle movement with either public transit or normal traffic flow, their main focus was on service vehicle traffic and other travel modes are modeled in a more simplistic way. 1.2. Contributions A majority of existing literature on multi-modal new mobility equilibrium resorts to simulation (Fagnant and Kockelman, 2014; Fagnant et al., 2016; Levin et al., 2017), partly because of complex interactions across travel modes and intertwining relations between supply and demand. This research targets to fill the gap by providing a theoretical framework of generic traffic network equilibria to unify these services and provides an analytical tool for prediction of impacts of various shared mobility models on traffic congestion. Its analytical features allow city planners to make optimal policy decisions for shared mobility services including transportation network design, pricing, and TNC regulation. A major difference of this paper from our previous papers on RUE (Di et al., 2017, 2018) is the integration of multiple modes including solo driving, self-organizing RS, and e-hailing, which adds considerable complexity into the mathematical framework. Generalizing traditional route choice based traffic equilibrium to RS primarily requires an extended network representation with generalized user equilibrium conditions, while an extension to further include e-hailing makes the overall framework a Nash game of the general type (Ban et al., 2018). This is because of the existence of both supply and demand side players and the coupling of their decisions. To disentangle such complexity, a super extended network representation is developed, built upon the extended network proposed in our previous research. The behavior and interactions of solo driving, e-haling, and RS into the traditional route choice based traffic equilibrium framework can be reflected and captured in this extended super network. This could help understand how the operations of these shared mobility on traffic congestion and the interaction or competition between these shared mobility modes. Integration of both RS and e-hailing into one modeling framework is crucial because they share certain resemblance but also exhibit subtle differences. A single-occupancy personal vehicle and a vacant e-hailing vehicle seem to contribute to traffic congestion equally. However, the former fulfills one unit of travel demand while the latter merely incurs deadhead miles. Similarly, a carpooling personal vehicle carries two units of demand while an occupied taxi (with one driver and one passenger) transports only one unit of demand. Therefore they contribute differently to traffic congestion. Another difference is that RS permits one passenger to be dropped off by one driver before reaching this passenger’s destination and to be picked up by another driver to continue the journey, while e-hailing vehicles are prohibited to drop off a passenger before reaching her destination. In conclusion, the ultimate goals of this research are: 1. to establish an analytical framework to quantify the interactions between personal vehicles and service fleets and the competition among different service providers; 2. to understand the impacts of the wide adoption of e-hailing on transportation system performance in terms of traffic congestion and deadhead miles incurred; 3. to investigate people’s mode choice in the presence of shared mobility services as travel cost and demand structure vary; 4. to evaluate the effectiveness of policies associated with ridesharing and e-hailing on the overall system performance. The anticipated contributions from this paper are: 1. It unifies mobility services into one framework wherein the congestion effects arising from the interactions of multiple mobility services can be modeled and evaluated; 2. It introduces a super extended network representation to model complex interactions among travel entities and a balance between travel demands and supplies. Mode choice and route choice are also modeled jointly in this representation; 3. The behavioral distinction in personal vehicle drivers and e-hailing drivers is captured in their associated vehicular flow networks, by introducing new links and nodes. Pesonal vehicles can swtich roles at an intermediate node by dropping off RS riders before reaching their destinations, while e-hailing vehicles can only switch states (occpuied or vacant) at origins. 4. Equilibrium solutions of the developed model reveal relations between travel modes, laying a theoretical foundation for network design problem that allows city planners to solve for optimal policy decisions for shared mobility services including transportation network design, pricing, and TNC regulation.

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The rest of the paper is organized as follows: in Section 3, the utility functions of available travel modes are defined and mode choice is modeled. In Section 4, route choice and flow conservation of ridesharing models are introduced. Section 5 describes the e-hailing vehicles’ operational model. 2. Preliminaries 2.1. Equilibrium modeling scheme Fig. (1) along with Table 1 illustrate the game structure and how the travelers and vehicles interact with each other at equilibrium. Travel needs of travelers moving from origins to destinations are the sources of all the games modeled in this paper and vehicular traffic on roads. Prior to a trip, one traveler selects travel mode with the minimium travel cost: driving her personal vehicle (PV), ridesharing a personal vehicle, or hailing an e-taxi. These pre-trip travel modes are denoted as M pre = {sr, r, em , em }, where sr denotes driving personal vehicles, r denotes becoming a RS rider, em denotes being an e-hailing passenger served by the mth provider, and em denotes the mth e-hailing service provider. Denote a set of e-hailing service providers as em ∈ Me . At each intermediate node, those who drive their personal vehicles can further decide to share rides with ridesharing (RS) riders for a part of or the entire journey, with the aim to minimize their generalized travel cost (i.e., travel cost minus RS earnings). RS riders can also choose which vehicles to take at individual intermediate nodes to minimize their generalized cost (i.e., travel cost plus RS payment). But the capacity constraint between RS drivers and riders needs to be perserved. The travel modes on roads thus include Men = {s, r, r, em , em }. For those passengers who choose e-hailing provider em ∈ Me , each platform needs to match vacant e-hailing vehicles with order requests, in order to maximize its total profit. After an vacant e-hailing vehicle is matched an order, it selects the fastest path to pick up the order. Once the vacant vehicle picks up an order and thus becomes an occupied vehicle, it aims also to move to the destination of the order along the fastest path. Note that the competition between e-hailing providers are not modeled directly. Instead e-hailing providers select their revenue structures first and leave it for travelers

Fig. 1. Game Structure. (Rounded rectangles represent players involved in the game: traveler (red) whose travel needs incur traffic; e-hailing platform (yellow) responds to e-hailing demand by matching vacant vehicles and orders. Six entities on road (green) are incurred by travelers’ needs. Ovals are game-play strategies). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 Game description. Player

Obj.

Strategy

Constraints

Traveler e-hailing platform

min. modal cost max. profit

driving PV, RS, e-hailing vehicle-order matching

vehicle flow ≥ order request

min. min. min. min. min. min.

role, route role, route RS vehicle mode route route

RS capacity Ehailing capacity

On road

Solo driver RS driver RS rider e-hailing passenger e-hailing driver occupied vacant

generalized route cost generalized route cost modal cost travel cost travel time to drop off travel time to pick up

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to decide which provider to take. Once the demand for each provider is determined, the role each platform plays is simply to match orders with vacant vehicles. On roads, the vehicular flow includes solo driving, ridesharing vehicles, and vacant or occupied e-hailing vehicles, which make route choices and thus contribute to traffic congestion. In return, the congestion cost affects modal cost and thus people’s mode choice. All the involved agents make decisions simultaneously not sequentially, therefore the game developed in this paper is a simultaneous game. Several assumptions are made throughout the paper: (A1) One unit of RS vehicle flow is allowed to carry at most Cap units of RS passenger flow, where Cap ≥ 1 is the capacity ratio of one unit of RS vehicular flow, i.e., the maximum units of riders carried by one unit of RS vehicular flow. (A2) One e-hailing vehicle is only allowed to take one passenger at a time. (A3) Personal vehicles and e-hailing vehicles are not allowed to switch their respective roles, in other words, a personal vehicle cannot become an e-hailing vehicle en-route and vice versa. (A4) Once a traveler decides not to drive, she can either become a RS rider or an e-haling passenger. By assumption, her role cannot be swapped en-route. In other words, transfer at intermediate nodes among multiple modes is not taken into account. (A5) RS service is self-organizing or ad-hoc. In other words, RS riders can be dropped off at an intermediate node and picked up by different RS vehicles. Note. 1. Assumption (A2) says that no pooling is considered for e-hailing service in this paper. The key challenge of incorporating pooling or multiple pickups for e-hailing service is to enumerate all the possible combinations of pickups and more importantly, their pickup ordering. If there are N total orders and 2 shared rides per trip, there are N (N − 1 )/2 combinations of orders; for each combination, there are 2 ordering possibilities. This is basically a permutation problem with time complexity of O(N2 ). A direct brute-force method is practically infeasible, not mentioning if 3 shared rides can be carried for one trip. We believe a quite different and innovative framework is required to accommodate pooling for e-hailing and we will leave it for future research. 2. Assumption (A3) says that there is no role switching between personal vehicle drivers and e-hailing drivers. In other words, at origin, a driver decides to either use her own vehicle as personal vehicle or as e-hailing vehicle and does not change it en-route. 3. Assumption (A5) may be violated when the e-hailing platform offers hitch service, wherein the e-hailing drivers provide RS while accomplishing their personal trips. In this case, RS service is commercialized and may be subject to restrictions such as not allowing drivers to drop off RS riders prior to riders’ destination. Then the RS model proposed in this paper has to be modified to impose these restrictions. Such revision is not straightforward and will be left for future research. The focus of this paper is to understand how personal and e-hailing vehicles contribute to traffic congestion, which lays a basic theoretical foundation for scenario variants. To avoid distracting from this focus, we will leave the relaxation of above assumptions for future research. 2.2. Super extended network representation To model the complex interactions of these entities and different types of traffic flows, a super extended network needs to be developed from a single-mode traffic network. A single-mode traffic network is represented by a directed graph G = {N , L} that includes a set of consecutively numbered nodes, N , and a set of consecutively numbered links, L. Let  denote the origin-destination (OD) pair set connected by a set of simple paths (composed of a sequence of distinct nodes), P ω , through the network. The notations regarding OD pairs can be expressed as one Greek letter ω or a pair of nodes (i, d ), ∀i ∈   NO , ∀d ∈ ND or (oω , dω ). In other words, ω = (i, d ) = (oω , dω ) are interchangeable. The node set can be divided into three subsets: the origin-node set denoted as NO , the intermediate-node set denoted as NI , and the destination-node set denoted as ND . Let us extract a subnetwork with i ∈ NO , d ∈ ND , j, k ∈ NI (in Fig. 2a), the structure of its super extended network is illustrated in Fig. 2b. The super extended network of an original network G, denoted as S (G ), is composed of four graphs separated in two layers (i.e., vehicular flow layer and person flow layer), plus virtual nodes and a set of virtual links which tie all travel modal networks at virtual nodes, coupled by inter-layer lines. The components of a super extended network are summarized in Table 2: 2.2.1. Vehicular flow layer The vehicular flow layer contains personal vehicle flow (sr) and e-hailing vehicle flow (ez). Other than their interactions on each link contributing to traffic congestion altogether, personal vehicles and e-hailing vehicles cannot switch their respective roles. In other words, a personal vehicle cannot become an e-hailing vehicle en-route and vice versa. To separate

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Fig. 2. Super extended network representation. Table 2 All components of a super extended network. Layer Vehicular flow Person flow Demand links

Inter-layer coupling

Graph

Notation

Node

Personal vehicle drivers E-haling drivers Ridesharing riders E-haling passengers Virtual origin nodes Virtual destination nodes Virtual links

G (sr )

G (ez ) G (r ) G (e )  NO  ND  L

N (sr ) = N N (ez ) = N N (r ) = N N (e ) = N  NO = NO  ND = ND –

Personal vehicle-RS E-hailing drivers-passengers

Lsr−r Lez−e

– –

Link  L (sr ) = L (s ) L (r ) L (ez ) = L (e ) ∪ L (z ) L (r ) = L L (e ) = L – –  L =   (i , i (sr )), (i0 , i (r )), (i0 , i (e ))} i0 ∈NO { 0  Lsr−r = k∈N {(k(sr ), k(r ))}  Lez−e = i∈NO {(i (ez ), i (e ))}

these two flows, the original network is made two copies to represent two networks: personal vehicle network (G (sr ) ) and e-hailing vehicle network (G (e ) ). In the personal vehicle network (G (sr ) ), links are doubled to represent solo driving vehicle and RS vehicle flows, respectively. On the other hand, solo and RS vehicles can switch roles en-route by picking up or dropping off passengers, therefore they share the same node set. Thus a personal vehicular network is composed of a node set N (sr ) and a link set  L(sr ) = L(s ) L(r ) where L(s ) and L(r ) are the exact copies of links in the original network. Then G (sr ) = {N (sr ), L(sr )}.

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Fig. 3. Occupied v.s. vacant e-hailing flows.

An e-hailing vehicle has two states: being occupied with a passenger or being vacant searching for the next passenger, illustrated in Fig. 3. E-hailing vehicles under both states contribute to traffic congestion. Thus, unlike personal vehicles that only generate traffic flows from origins to destinations, e-hailing vehicles incur two types of flows: OD flow and DO flow. OD flow, incurred by travelers demand for this mode, is occupied e-hailing vehicle flow connecting the origin to the destination of a trip; while DO flow, arising from the need to pick up an e-hailing order, is vacant vehicle flow connecting the destination of a previous order and the origin of the next pickup order. Denote the DO pair set as   {ω : ω = (dω , oω ), ∀ω ∈ }. A major difference between not-for-profit RS and e-hailing is that RS permits one passenger to be dropped off by one driver before reaching this passenger’s destination and to be picked up by another driver to continue the journey, while e-hailing vehicles are prohibited to drop off a passenger before reaching her destination. To accommodate such restriction, in the e-hailing vehicle network (G (ez ) ), the link set L is made two copies connecting the same node set: the link set where occupied e-hailing vehicle flow is on, denoted as L(e ), and that where the vacant e-hailing vehicle flow is on, denoted as L(z ). Then L(ez ) = L(e ) ∪ L(z ). d (e ) o(e ) Denote xi j m as the e-hailing vehicle flow on link (i, j) to destination d and xi j m as the e-hailing vehicle flow on link (i, j) to origin o. Traffic flows in the vehicular flow layer contribute to traffic congestion. Define xij as the aggregate vehicular link flow on      d ( em ) link (i, j) where xi j = d∈N  xdi j(s ) + xdi j(r ) + d∈N  ∪N . Then x is the aggregate vehicular link flow vector. Let m∈M xi j D

D

O

t(xij ) be the link travel time function of link ij, which is a monotonically increasing function of the vehicular link flow and depends on the aggregate vehicular link flow, i.e., ti j = t (xi j ). Then Tp (f ) = T t (x ) under the additive path cost assumption. 2.2.2. Person flow layer The person flow layer contains RS rider flow (r) and e-hailing passenger flow (e). Once a traveler decides not to drive, she can either become a RS rider or an e-haling passenger. By assumption, her role cannot be swapped en-route. To separate these two flows, the original network is made two identical copies: RS rider network (G (r ) ) and e-hailing passenger network (G (e) ). In the RS rider network (G (r ) ), riders can only be riders throughout the network. The rider network is composed of a node set N (r ) and a link set L(r ), i.e., G (r ) = {N (r ), L(r )}. Riders do not need to decide which route to take but rather follow drivers. However, the number of RS riders on each link is constrained by the total seat capacity of RS vehicles on that link. In other words, the rider link flow is determined by the corresponding personal vehicular flow. Through capacity constraints, the personal vehicle network and RS rider network are coupled. To represent such coupling across layers, each node in the personal vehicle network is connected to its corresponding nodes in the RS passenger network using dotted lines. These lines represent the following mathematical coupling relation: (r )

xi j − xi(jr )  0, (r )

Cap · xi(jr ) − xi j  0.

(2.1a) (2.1b)

The e-hailing passenger flow (e) does not generate but incurs e-hailing vehicular flow (e). Accordingly the e-hailing passenger network (G (e) ) and e-hailing vehicle network (G (e ) ) are coupled via supply constraints. In other words, vacant ehailing vehicles have to satisfy the total demand and their spatial distribution determines e-hailing passengers’ waiting cost. Nevertheless, because the above coupling relations only happen at the origin nodes of travel demands, these two networks are mainly linked at origin nodes, different from those coupling lines tying RS networks. Among all four graphs, there exist three travel modal networks, which are personal vehicle network (G (sr ) ), RS rider network (G (r ) ), and e-hailing passenger network (G (e) ). To ensure the conservation of traffic demands, one virtual origin node ∀i ∈ NO and one virtual destination node ∀d ∈ ND are created to represent a source node and a destination node of the super extended network. Three travel modal networks (i.e, personal vehicle network (G (sr ) ), RS rider network (G (r ) ), and e-hailing passenger network (G (e) )) are connected by virtual links from virtual source nodes to the origin of each network and from a destination of each network to virtual destination nodes. The virtual links connecting virtual source nodes to   G (sr ) , G (r ) , G (e) are denoted as L (sr ), L (r ), L (e ), respectively, where L (sr ) = i∈N  (i, i(sr )), L (r ) = i∈N  (i, i(r )), L (e ) = O O   (i, i (e )). The travel cost on virtual links are assumed to be zero so that adding these links will not alter people’s mode i∈N O

choice behavior as they do in the original network.

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3. Travelers’ mode choice model 3.1. Traveler flow How does a traveler flow through this super network? At the virtual origin node i, a traveler faces three mode choices: driving a personal vehicle, riding a personal vehicle, or e-hailing. If she decides to drive, this traveler moves to the personal vehicle network (G (sr ) ) via the virtual link (i, i(sr)). Within the personal vehicle network, the traveler can switch between solo and RS by picking up or dropping off passengers at each node. If she decides to carpool with other travelers, this traveler moves to the RS rider network (G (r ) ) via the virtual link (i, i(r)). Within the RS rider network, she cannot choose routes but rather follow drivers. So dotted lines linking the RS rider network to the personal vehicle network represent capacity constraints of RS vehicles on each link. If she decides to take e-hailing service, this traveler moves to the e-hailing passenger network (G (e) ) via the virtual link (i, i(e)). Within the e-hailing passenger network, similarly, she cannot choose routes but rather follow drivers. Her waiting time and matching cost is determined by the available e-hailing vehicles (G (e ) ). Such coupling between the e-hailing passenger network and the e-hailing driver network is represented by the dotted line connecting their respective origins.

3.2. Demand conservation A traveler faces three pre-trip mode choices: driving a personal vehicle, riding a personal vehicle, or e-hailing. Denote the traffic demand between OD pair ω = (i, d ) as qdi > 0. Accordingly the total travel demand qω is split into three d (r )

modes, denoted as qdi (sr ) , qi

d (e )

, qi

d (em )

, respectively. qi d (e ) q i m , ∀m

can be further decomposed into different e-hailing service providers d (e )

em , ∀m ∈ M. Subsequently we will use ∈ M to replace qi for e-hailing. For the fixed travel demand case, the demand conservation condition needs to be satisfied at each virtual origin node corresponding to πid ≥ 0:



0



 j:(i, j )∈L

qdi (sr )

+

d (r ) qi

+

d (e ) qi





−

qdi

⊥ πid  0, ∀i ∈ N  , d ∈ ND .

(3.1)

3.3. Modal cost Before delving into choice model, the notations of costs are listed in Table 3. The travel disutility of being solo drivers and RS drivers for OD pair ω = (i, d ) is the sum of traffic congestion cost, inconvenience cost, and revenue: d (r ) d (sr ) Cid (sr ) = Fi(sr ) + Cid (sr ) + Cinc,i − C p,i .







Cost

Earning

(3.2)

Table 3 Notations associated with cost. Mode

Description

Modal Cost

Link Cost Parameter

Driving PV

sr

Solo driver RS driver RS rider RS

s r r r, r

EPassenger

em

EDriver

em

Link

Fixed cost representing car depreciation and parking cost Time- and distance-based cost Time- and distance-based cost Time- and distance-based cost Time-based cost Inconvenience cost RS Earning/Fee Time-based cost Waiting cost Fixed fare charged at origin Time- and distance-based fare Time- and distance-based cost Length Free-flow travel time flow

Nodal Cost Cost Fi(sr )

Cid (sr )

tid d (r ) d (r ) Cinc,i , Cinc,i

d (r )

d (sr ) , C p,i C p,i

tid

β1(s) , β2(s) β1(r ) , β2(r ) β1(r ) (r )

(r ) Iinc , Iinc

(r )

I p(r ) , I p

β1(em ) β2(r ) , β3(r )

tid − t¯id , lid tid , lid

α1(em ) , α2(em ) β1(em ) , β2(em )

lid lid

lij t¯i j xij

tij (x), lij (x) tij (x), lij (x) tij (x) d (r ) d (r ) cinc,i , cinc,i j j d (r )

cdp,i(rj) , c p,i j tij (x)

cw, i Fi(em ) tij (x), lij

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The travel disutility of being RS riders for OD pair ω is the sum of traffic congestion cost, inconvenience cost, and RS fee: d (r )

Ci

(r )

d (r )

d (r )





= β1 tid + Cinc,i + C p,i





Cost

.

(3.3)

Payment

The travel disutility of being e-hailing passengers is the sum of fixed cost, distance-based, and time-based cost, and waiting cost: d (em )

Ci



( em ) d

= − Fi(em ) + α1(em ) (tid − t¯id ) + α2(em ) lid + β1







F are

ti + cw,i .





(3.4)

Cost

3.4. Mode choice 

At node ∀i ∈ NO , travelers choose which virtual link to take: taking link (i, i(sr)) leads to driving personal vehicles, taking link (i, i(r)) leads to RS riders, and taking link (i, i(e)) leads to taking e-hailing service. Therefore, in the super extended network, the mode choice can be modeled as route choice along virtual links.    Define πid , ∀i ∈ NO as node potentials for the virtual source node i ∈ NO to the virtual destination node d ∈ ND in the super extended network. The complementarity of mode choice is:



0  Cid (sr ) − πid (sr ) ⊥ qdi (sr )  0, ∀ω ∈ 



d (r )

0  Ci



d ( em )

0  Ci

d (r )

− πi



d (em )

− πi

d (r )

⊥ qi



 0, ∀ω ∈ 

d ( em )

⊥ qi

 0 , ∀ ω ∈ , e m ∈ M e .

(3.5a) (3.5b) (3.5c)

where ⊥ is the orthogonal sign representing the inner product of two vectors. At equilibrium, no traveler (i.e., personal vehicle driver, RS rider, e-hailing passenger) can reduce her generalized modal disutility by unilaterally switching travel modes pre-trip, i.e., the node potential of utilized mode is equal, less than that of unused modes, as long as the cost on virtual links are set to zero:

qdi (m )



> 0 ⇒ Cid (m ) = πid (m ) = 0 ⇒ Cid (m ) > πid (m )

, ∀m ∈ M pre .

Note. Mode choice can also be described using logit model or its variants (Wong et al., 2001a; Yang et al., 2002, 2010a, 2014; Wong et al., 2008; He and Shen, 2015; Qian and Ukkusuri, 2017), assuming that there exist perception errors in travel cost. In this paper, the deterministic modal choice is assumed to make consistent assumptions of the users’ perfect perception of costs in mode choice and route choice. If we were to assume perception errors on the mode choice costs, i.e., stochastic modal choice, we would also need to consider perception errors for route choice. In other words, stochastic user equilibrium instead of deterministic user equilibrium needs to be applied. This could dramatically increase the complexity of the model, in terms of both analysis and computation. The scope of this paper is to provide a fundamental framework to incorporate several shared mobility. For this, we choose a deterministic framework to start with. The extension to stochastic versions will be left for future work. 4. Ridesharing model in the link-node representation The extended network representation to model RS flows was originally proposed by Xu et al. (2015) but was articulated and extended to the link-node formulation by Di et al. (2018). To facilitate the modeling formulation of role switching between solo and RS drivers at an intermediate node, solo and RS vehicle flows share the same node set but travel on different link sets (shown in Fig. (2)). RS driver flows in a separate network. The travel cost for each vehicle type is computed differently, while the sum of two vehicle flows is conserved at each intermediate node. Such trick enables the role switching between solo and RS drivers at an intermediate node. On each link, RS driver and rider flows satisfy capacity constraints. Subsequently we will introduce how route choice, RS vehicle capacity, and flow conservation are formulated. The travel cost for each type of traffic flow on link (i, j) is defined as follows (Xu et al., 2015; Di et al., 2018):

⎧ (s ) c (x ) = β1(s ) ti j (x ), solo drivers ⎪ ⎪     ⎨ i(jr ) (r ) (r ) (r ) (r ) r) ci j (x ) = β1(r ) ti j (x ) + Iinc cinc,i j xi(jr ) , xi j − I p(r ) c(p,i xi(jr ) , xi j , pi j , ridesharing drivers j   ⎪   ⎪ ⎩c(r ) (x ) = β (r )ti j (x ) + I (r ) c(r ) x(r ) , x(r ) + Ip(r ) c(r ) pi j , riders 1 ij inc inc,i j ij ij p,i j

where,

(4.1)

X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78





59

(r )

x (r ) (r ) Let cinc,i xi(jr ) , xi j = li j i(jr ) , i.e., the inconvenience cost increases when the number of ridesharing passengers increases j xi j

or the number of drivers decreases; (r )







(r )



r) Let c p,i j pi j = pi j , c (p,i xi(jr ) , xi j , pi j = pi j j

λ+i j , λ−i j

(r ) xi j (r ) , xi j

where pij is a constant premium for sharing rides on link ij.

Denote ≥ 0 as the multipliers for capacity constraints in Eqs. (2.1a)–(2.1b), respectively. The link costs in Eq. (4.1) can be converted to the generalized link costs:

⎧ (s ) (s ) ⎪ ⎨c˜i j = ci j ,   c˜i(jr ) = ci(jr ) + λ+ − Cap · λ− , ij ij ⎪ ⎩c˜(r ) = c(r ) − λ+ − λ− . ij ij ij ij

(4.2)

The ridesharing user equilibrium (RUE) satisfies the Wardrop first principle in terms of generalized travel costs, not the actual travel cost (Di et al., 2018). In other words, travelers take the routes with the same and minimum generalized travel cost rather than those with the minimum travel time, because it violates Cartesian product structure (Larsson and Patriksson, 1999). This equilibrium condition is similar to that for general user equilibrium with side constraints developed in Larsson and Patriksson (1999). Accordingly, at ridesharing user equilibrium, for each OD pair ω where ω ∈ ⊗, no traveler (i.e., solo driver, RS driver, RS rider) can improve her generalized travel cost by either unilaterally switching routes or roles. Mathematically, (m )



> 0 ⇒ C˜p(m ) = πid (m )

fp

, ∀ p ∈ Pid , ∀m ∈ M.

= 0 ⇒ C˜p(m ) > πid (m )

where f p(m ) is the path flow for travel mode m = {s, r }, and πid (m ) ≥ 0 is the minimum generalized path cost for OD pair (i, d) and travel mode m. 4.1. Route choice Route choice behavior for drivers and passengers are different as they experience different travel cost and has to be defined separately. However, as solo and ridesharing drivers share the same vehicular network, only one set of node potentials is needed for the vehicular network. Define πid(sr ) ≥ 0, ∀i ∈ N (sr ) and πid(r ) ≥ 0, ∀i ∈ N (r ) as node potentials for nodes in the vehicular traffic and passenger networks, respectively. Then the link-node based complementarity conditions related to route choice are applied to links within the vehicular network and the passenger network respectively:

0 0 0

  π jd(sr ) + c˜i(js) (x ) − πid(sr ) ⊥ xi(js)  0, ∀(i, j ) ∈ L(sr ), i, j ∈ N (sr ), d ∈ ND ,

(4.4a)

  π jd(sr ) + c˜i(jr ) (x ) − πid(sr ) ⊥ xi(jr )  0, ∀(i, j ) ∈ L(sr ), i, j ∈ N (sr ), d ∈ ND ,

(4.4b)



  π jd(r ) + c˜i(jr ) (x ) − πid(r ) ⊥ xi(jr )  0, ∀(i, j ) ∈ L(r ), i, j ∈ N (r ), d ∈ ND .

(4.4c)

4.2. Ridesharing flow capacity constraints According to the capacity constraints (2.1), define the occupancy ratio oij for link (i, j) as: (r )

xi j oi j = ( r ) . xi j

(4.5)

Then oij is bounded below by one and bounded above by the capacity ratio, i.e.,

oi j  1 ,

(4.6a)

oi j  Cap.

(4.6b)

Note. The occupancy of one link does not have to be one nor Cap, it can be a continuous number between 1 and Cap. It is not a predefined number, but rather intrinsically determined by the equilibrium. The denominator xi(jr ) in the definition of the occupancy ratio can be zero if the ridesharing driver flow is zero, which causes the cost to be indefinite. To avoid 00 , we will introduce oij into the complementarity formulation as a variable. The occupancy ratio oi j , ∀(i, j ) ∈ L can be treated as the optimal solution of the following quadratic program:

oi j = arg min

o∈[1,Cap]

xi(jr ) 2

(r )

o2 − xi j o.

(4.7)

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X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

This trick, also employed in our previous study (Di et al., 2018), is a standard technique used to avoid zero denominator. The rationale is: for the scalar quadratic program (4.7) with respect to the decision variable o, constrained by Eqs. (4.6a)– (4.6b), the minimum is always attained at the lowest point of the parabola, i.e., oi j =

(r ) xi j (r ) , xi j

which is Eq. (6.3d). Therefore the

quadratic program (4.7) is equivalent to (6.3d–4.6). o Denote λ+ , λ−o as the multipliers for capacity constraints in Eqs. (4.6a)–(4.6b), respectively. The KKT conditions of the ij ij quadratic program (4.7) can be characterized by the following complementarity conditions: (r )

0  oi j ⊥ xi(jr ) oi j − xi j + λo+ − λo−  0, ∀(i, j ) ∈ L(sr ), i, j ∈ N (sr ), d ∈ ND , ij ij

(4.8a)

0  λo+ ⊥ oi j − 1  0, ∀(i, j ) ∈ L, ij

(4.8b)

0  λo− ⊥ Cap − oi j  0, ∀(i, j ) ∈ L. ij

(4.8c)

4.3. Flow conservation Now we will define ‘flow conservation’ on the copied nodes in vehicular and passenger networks respectively and ‘demand conservation’ on virtual nodes connecting these two networks. Note that source nodes in the original network become intermediate nodes in vehicular and passenger networks. Flows d (r ) should be conserved at all the nodes in each copied network corresponding to πid (sr ) , ∀i ∈ N (sr ) and πi , ∀i ∈ N (r ), respectively:



0





 j:(i, j )∈L(sr )



0

xdi j(s )

d (r )

j:(i, j )∈L(r )

xi j

+

−

xdi j(r )



−

 k:(k,i )∈L(r )



  k:(k,i )∈L(sr ) L (sr )

xdki(s )

(r )

+ xki





−

qdi (sr )

⊥ πid (sr )  0, ∀i ∈ N (sr ), d ∈ ND , 

(4.9a)



d (r )

xki

d (r )

− qi

d (r )

⊥ πi



 0, ∀i ∈ N ( r ), d ∈ ND .

(4.9b)

Eq. (4.9a) says that solo and RS driver flows are conserved at each intermediate node, but they can switch roles across links. 5. E-hailing operational model in the link-node representation Table 4 defines two types of e-hailing vehicle flows. Given the assumption that one e-hailing vehicle is only allowed to d (e ) take one passenger, the occupied e-hailing vehicle flow equals to the OD demand qi m , ∀(i, d ) ∈ . The travel behavior of occupied e-hailing vehicles is similar to that of personal vehicles: to transport passengers using the fastest route. At equilibrium, no occupied vehicle can improve its travel time by unilaterally switching routes. Unlike occupied vehicles, vacant e-hailing vehicles need to be first matched to the next order. Once the next served orders are determined, vacant vehicles take a fastest route to pick up passengers. In other words, this is a two-step process: vacant vehicle assignment and route choice. Subsequently we will first model vacant vehicle and order maching and then develop a route choice model for vehicles in both states. 5.1. E-hailing platform operational choice On an e-hailing platform, drivers do not have discretion to decide whom to pick up. Instead, such choices are jointly optimized by the e-hailing platform, with the goal of maximizing the total profit of all drivers using this platform during a designated time period. In other words, order matching is a centralized optimization, not a noncooperative game. d  ( em )

Denote zi ω

as the decision variable that the e-hailing platform m ∈ Me made at destination i ∈ ND as to how many d  ( em )

vehicles will be dispatched to dω = oω , ∀ω ∈  , ω ∈  to pick up passengers of OD pair ω. We also call zi ω the DO flow as it is the total number of vehicles connecting the destinations of the previously served OD pair to the origins of

Table 4 Two types of e-hailing vehicle flows. Vehicle flow

State

Description

OD flow DO flow

Occupied Vacant

connecting ω ’s origin to its destination, i = oω , j = dω connecting ω ’s destination to ω ’s origin, i = dω , j = oω

Quantity  d (e ) qi m e m∈M ω ∈

m∈Me

zidω (em )

X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

61

( em ) the next served OD pairs. The net profit per ride earned at destination i to serve ω, denoted as Rω , is computed as the i revenue minus the travel cost ∀(i, ω ) ∈ ND × : ( em ) Rω = Fi(em ) + α1(em ) (todωω − t¯odωω ) + i







time−based





Revenue

α2(em ) lodωω − β1(em ) (tioω + todωω ) − β2(em ) (lioω + lodωω ) . 



 distance−based time−based distance−based



(5.1)

Cost

The profit earned at destination i to serve ω by e-hailing platform m is computed as the product of a single ride’s net d  ( em )

profit and the dispatched DO flow zi ω . Then the total profit for the platform m is the sum of the profit across all drop-off   ( em ) dω  ( em ) nodes and travel demand OD pairs: ω∈ i∈ND Rω zi . i d  ( em )

There are two constraints regarding the DO flow zi ω . The first constraint is that the available vacant vehicle flow at node i must be equal to the total vehicle flow that serves the travel demands ending at the destination node i, i.e.,   dω  ( em ) i (e ) = ω∈ qoω m , ∀i ∈ ND . The second constraint is that the vacant vehicle flow to serve OD pair ω needs to be ω ∈ zi  o  ( em ) d  ( em ) d (e ) greater than or equal to the travel demand of OD pair ω, i.e., i∈ND zi ω  qoωω m , ∀ω ∈ . Because we assume zi ω = zioω (em ) , ∀ω ∈  , ω ∈ , we will use zioω (em ) from now on to avoid confusion. In summary, the e-hailing platform needs to solve the following optimization to match vacant vehicle flows with passenger requests:

max

o (e ) zi ω m 0

ω∈ i∈ND

s.t.  

  ω (e ) o (e ) Ri m zi ω m ,

ω ∈

zioω (em ) =

 ω ∈

d (em )

zioω (em )  qoωω

i (e )

qoω m , ∀i ∈ ND

, ∀ ω ∈ .

(5.2a) (5.2b)

i∈ND

Define the multipliers associated with constraints (5.2a)–(5.2b) as φ i and λω (em ) ≥ 0, respectively. Then the above optimization program can be converted to the following nonlinear complementarity problem (NCP): ( em ) 0  zioω (em ) ⊥ −Rω − φi − λω (em )  0, ∀(i, ω ) ∈ ND × , i

 ω ∈

0

zioω (em ) =  i∈N 

 ω ∈

i (e )

qoω m , φi free, ∀i ∈ ND , d (em )

zioω (em ) − qoωω

⊥ λω (em )  0, ∀ω ∈ .

(5.3a) (5.3b)

(5.3c)

D

Note. The e-hailing service providers are in a generalized Nash game wherein the constraints of one player may involve the decision variables of other players. In other words, one e-hailing service providers strategy (i.e., matching vacant vehicles and orders) does not directly enter others profit functions. Instead, their admissible strategy sets (i.e., dispatch of feasible vacant vehicle flows) are coupled through total travel demand. This makes the proposed framework mathematically complex. After vacant vehicles know where to pick up the next passenger, they either choose routes on their own or follow navigation guidance from the platform, with the same goal to take the fastest route to the pick-up nodes. 5.2. Link-node based e-hailing route choice OD and DO flows share the same e-hailing vehicular network and contribute to traffic congestion jointly. However, vehicle states cannot be switched at intermediate nodes, except at origin or destination nodes: only at a destination node, an occupied vehicle becomes a vacant one after a drop-off; similarly, only at an origin, a vacant vehicle becomes an occupied one after a pickup. Accordingly, we have to separate OD and DO flows in flow conservation. Then route choice has also to be defined respectively for these two types of flows. Di et al. (2018) proposed two approaches to represent two parallel flows connecting the same node pair: one is to simply introduce a parallel link with different link flow notation, while the other is to add a dummy node to break the second parallel link without new notations. Both approaches have pros and cons but do not alter properties of the original problem. We adopt the first one here, i.e., introducing two link flow notations, illustrated in Fig. 4.   Define xdi j(e ) = m∈Me xdi j(em ) as the occupied flow on link (i, j) to an OD flow destination d ∈ ND , and xdi j(z ) = m∈Me xdi j(zm )

as the vacant flow on link (i, j) to a DO flow origin d ∈ NO . Accordingly, define τid (e ) , τid (z ) ≥ 0 as node potentials of the intermediate node i ∈ N (ez ) associated with OD and DO flows, respectively (Fig. 4).

62

X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

Fig. 4. Link-node based e-hailing route choice.

The complementarity conditions of route choice for OD and DO flows are:

0  xdi j(e ) ⊥ τ jd (e ) + ti j (x ) − τid (e )  0, ∀(i, j ) ∈ L(e ), i, j ∈ N (ez ), d ∈ ND ,

(5.4a)

0  xdi j(z ) ⊥ τ jd (z ) + ti j (x ) − τid (z )  0, ∀(i, j ) ∈ L(z ), i, j ∈ N (ez ), d ∈ NO .

(5.4b)

The complementarity conditions of flow conservation for OD and DO flows are:





0



j:(i, j )∈L(e )



0

j:(i, j )∈L(z )

xdi j(e )

xdi j(z )

−

 k:(k,i )∈L(e )

−

 k:(k,i )∈L(z )

xdki(e )

xdki(z )

−





d (e ) qi m

⊥ τid (e )  0, ∀i ∈ N (ez ), d ∈ ND ,

m∈Me

−

  ω∈ m∈Me

(5.5a)

 zidω (em )

⊥ τid (z )  0, ∀i ∈ N (ez ), d ∈ NO .

(5.5b)

At user equilibrium, for each OD pair ω ∈  or DO pair ω ∈  , no e-hailing driver can reduce her travel time by unilaterally switching routes. Mathematically, (m )

fp



> 0 ⇒ t p(m ) = τid (m ) = 0 ⇒ t p(m ) > τid (m )

, ∀ p ∈ Pid , ∀m = {e, z},

where f p(m ) is the path flow for e-hailing vehicles, t p(m ) is the path travel time, and τid (m ) ≥ 0 is the minimum travel time for OD pair (i, d) and travel mode m. 5.3. Customers’ waiting cost For those passengers who choose to take e-hailing servie, their waiting cost coω is composed of two parts: the vacant vehicles’ travel time from the last drop-off node to the pick-up node, and the matching cost which is inversely proportional to the number of vacant vehicles. Vacant vehicles’ average travel time to the origin of OD pair ω from a drop-off node k ∈ ND can be computed as: 

oω ( em ) oω t k∈ND zk k oω ( em )  zk k∈N D



.

The matching cost is characterized by the multiplier λω (em ) associated with the constraint defined in Eq. (5.2b). ∀ω ∈ ,



d (em )

⇒ λω (em ) = 0, (passengers no waiting)

(5.7a)

d (em )

⇒ λω (em ) > 0. (passengers waiting)

(5.7b)

zioω (em ) > qoωω

i∈ND



zioω (em ) = qoωω

i∈ND

In case (5.7a) when there is a surplus of e-hailing vehicles, passengers do not need to wait; while in case (5.7b) when there is no surplus of e-hailing vehicles, passengers have to wait and the waiting cost is proportional to the multiplier λω . Accordingly the multiplier λω can be treated as a fee an e-hailing passenger has to pay to drivers to ensure a shorter waiting time. It can be deemed as “demand price” or “surge price”. Surge price is widely used by e-hailing companies to balance out the supply and the demand in the real-world practice. When the demand is higher than the supply, a positive fee is paid by passengers to drivers to guarantee a timely service. In other words, the e-hailing system is a bilateral economic market

X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

63

where drivers are the supply side providing service while passengers are the demand side seeking service. The numbers of drivers and passengers depend on the trip cost. When the demand price is zero, there will be a surplus of vehicles. When the demand price is positive, fewer passengers will take the service. Note. 1. Only customers’ waiting time is considered, while e-hailing vehicles’ waiting time is not modeled. E-hailing drivers are dispatched by a platform whose goal is to maximize the total net profit. That being said, individual drivers’ waiting cost and next go-to-passenger choice are not modeled. This is different from the traditional taxi models (Wong et al., 20 01b, 20 08; Yang et al., 2010b; Yang and Yang, 2011; Yang et al., 2014) where individual taxi drivers’ choice is driven by their individual goals of maximizing a net profit with waiting and fuel consumptions. A matching function with a value of the number of matched pairs is proportional to the number of vacant vehicles and passengers, while the waiting time is inversely proportional to the matching value. To simplify, we assume there are a large amount of e-hailing drivers, therefore their total working hours are not posed as a constraint. This assumption also holds when shared autonomous fleet is modeled. An extension to include drivers’ waiting time, working hours, or fleet size will be left for future work. 2. To the best of our knowledge, all the existing literature on taxi drivers’ network models (Wong et al., 2001b, 2008; Yang et al., 2010b; Yang and Yang, 2011; Yang et al., 2014) assume idle drivers follow the same route choice behavior as the occupied vehicles, i.e., take the shortest paths and no vacant vehicles can improve travel time by unilaterally switching routes. Such assumption is also adopted in this paper. Idle drivers’ passenger-searching behavior can be quite complicated. There exist extensive studies on idle driver behavioral modeling (Liu et al., 2010; Powell et al., 2011; Li et al., 2011; Szeto et al., 2013; Sirisoma et al., 2010; Wong et al., 2014a, 2014b, 2015a, 2015b; Hu et al., 2012; Tang et al., 2016; Shou and Di, 2019), indicating that most time idle drivers do not follow the shortest path to pick up the next passenger, because they do not know right away where the next passenger is. A more general route choice behavior for vacant vehicles will be left for future research. In summary, the waiting cost can be written as: ( em )

cw,oω = β2



zkoω (em )tkoω

k∈ND



oω ( em ) k∈ND zk

(em ) ω (em )

+ β3

λ

, ∀ k ∈ N D , ω ∈ .

(5.8)

Note. In the waiting cost, no matching friction is considered as done in Yang et al. (2010b); Yang and Yang (2011); Yang et al. (2014), because we assume there does not exist any search friction on an e-hailing platform. To avoid the denominator in the definition of the vacant vehicles’ travel time to be zero, we use the similar trick in (4.7), by introducing θkω ≥ 0 that solves the optimal solution of the following quadratic program:



θkω ≡ arg min θ ∈ [0 , 1 ]

k ∈ND



k∈ND

zkoω (em )

zkoω (em )



θ − 2

zkoω (em )

θ .

(5.9)

Then the waiting cost can be rewritten as: ( em )

cw,oω = β2



k∈ND

θkω tkoω + β3(em ) λω(em ) , ∀k ∈ ND , ω ∈ .

(5.10)

Its complementarity conditions are:



0  θkω ⊥



k ∈N



zkoω (em )

θkω − zkoω (em ) + ζkω  0, ∀k ∈ ND , ω ∈ ,

(5.11a)

D

0  ζkω (em ) ⊥ 1 − θkω  0, ∀k ∈ ND , ω ∈ .

(5.11b)

Accordingly, the travel disutility of being e-hailing passengers defined in Eq. (3.4) is rewritten as: d (em )

Ci



(em ) d

= − Fi(em ) + α1(em ) (tid − t¯id ) + α2(em ) lid + β1

(em )

ti + β2



θki tki + β3(em ) λdi (em ) .

(5.12)

k∈ND

6. Mixed ad-hoc carpooling and e-hailing model Summarizing the above models, the mixed path and link-node based RUE (i.e., [RUE-NCP]) can be formulated as follows: Traveler mode choice (i.e., demand side):



0  Cid (sr ) − πid (sr ) ⊥ qdi (sr )  0, ∀ω ∈ 



d (r )

0  Ci

d (r )

− πi



d (r )

⊥ qi

 0, ∀ω ∈ 

(6.1a) (6.1b)

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X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78



d ( em )

0  Ci



d (em )

− πi





0

j:(i, j )∈L



d ( em )

⊥ qi

d (r )

qdi (sr ) + qi

 0 , ∀ ω ∈ , e m ∈ M e , d (e )

+ qi



(6.1c)

 − qdi

⊥ πid  0, ∀i ∈ N  , d ∈ ND .

(6.1d)

E-hailing passengers’ waiting cost (i.e., market price clearance):



0  θkω ⊥



k ∈ND



zkoω (em )

θkω − zkoω (em ) + ζkω  0, ∀k ∈ ND , ω ∈ ,

(6.2a)

0  ζkω (em ) ⊥ 1 − θkω  0, ∀k ∈ ND , ω ∈ ,

(6.2b)

RS route choice:

0 0 0

  π jd(sr ) + c˜i(js) (x ) − πid(sr ) ⊥ xi(js)  0, ∀(i, j ) ∈ L(sr ), i, j ∈ N (sr ), d ∈ ND ,

(6.3a)

  π jd(sr ) + c˜i(jr ) (x ) − πid(sr ) ⊥ xi(jr )  0, ∀(i, j ) ∈ L(sr ), i, j ∈ N (sr ), d ∈ ND ,

(6.3b)

   π jd(r ) + c˜i(jr ) (x ) − πid(r ) ⊥ xi(jr )  0, ∀(i, j ) ∈ L(r ), i, j ∈ N (r ), d ∈ ND ,

(6.3c)

(r )

0  oi j ⊥ xi(jr ) oi j − xi j + λo+ − λo−  0, ∀(i, j ) ∈ L(sr ), i, j ∈ N (sr ), d ∈ ND , ij ij

(6.3d)

0  λo+ ⊥ oi j − 1  0, ∀(i, j ) ∈ L, ij

(6.3e)

0  λo− ⊥ Cap − oi j  0, ∀(i, j ) ∈ L, ij

(6.3f)

 0





 j:(i, j )∈L(sr )



0

j:(i, j )∈L(r )

 d (r )

xdi j(s ) + xi j

d (r ) xi j

−



−

k:(k,i )∈L(r )



  k:(k,i )∈L(sr ) L (sr )

 (r )

xdki(s ) + xki

 − qdi (sr )

⊥ πid (sr )  0, ∀i ∈ N (sr ), d ∈ ND , 

(6.3g)



d (r ) xki

−

d (r ) qi

d (r )

⊥ πi



 0, ∀i ∈ N ( r ), d ∈ ND .

(6.3h)

E-hailing vehicle supply: ( em ) 0  zioω (em ) ⊥ −Rω − φi − λω (em )  0, ∀(i, ω ) ∈ ND × , i

 ω ∈

0

zioω (em ) = 

 ω ∈

i (e )

qoω m , φi free, ∀i ∈ ND , d (em )

zioω (em ) − qoωω

(6.4a) (6.4b)

⊥ λω (em )  0, ∀ω ∈ .

(6.4c)

i∈ND

E-hailing route choice:

0  xdi j(e ) ⊥ τ jd (e ) + ti j (x ) − τid (e )  0, ∀(i, j ) ∈ L(e ), i, j ∈ N (ez ), d ∈ ND ,

(6.5a)

0  xdi j(z ) ⊥ τ jd (z ) + ti j (x ) − τid (z )  0, ∀(i, j ) ∈ L(z ), i, j ∈ N (ez ), d ∈ NO ,

(6.5b)

 0

 0

 j:(i, j )∈L(e )

 j:(i, j )∈L(z )

xdi j(e )

xdi j(z )

−

 k:(k,i )∈L(e )

−

 k:(k,i )∈L(z )

xdki(e )

xdki(z )

−



 d (e ) qi m

m∈Me

−

  ω∈ m∈Me

⊥ τid(e )  0, ∀i ∈ N (ez ), d ∈ ND ,

(6.5c)

 zidω (em )

⊥ τid(z )  0, ∀i ∈ N (ez ), d ∈ NO .

The equilibrium of the new mobility equilibrium is a tuple (q, x, z, t, θ , o).

(6.5d)

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65

6.1. Game-theoretic model structure To illustrate the structure of the developed model, we first divide all the variables into primary variables and multipliers, respectively,

• •

Primary variables: v = q

 Multipliers: πid (sr )

πid (r )

x

z

πid (e)

t

θ τid (e)

T

o

;

τid (z)

φi

ζiω

λo+ ij

T . λo− ij

We also group nonlinear complementarity conditions into four modules, which are connected to one another through coupling constraints and variables. These four modules are demonstrated in Fig. (5). Within each module, we move the primary variables (colored in red) associated with these nonlinear inequalities to the left of the NCPs. The linear complementaries are rewritten as linear constraints. The multipliers for these linear constraints are colored in green. Module (A) contains Eq. (3.5) where the travel disutility for each mode is computed by Eqs. (3.2), (3.3), (5.12), respectively. This module depicts travelers’ modal choices at the origin node. The outputs are travel modal demands, which are fed into Module (B) and (D) as constraints, respectively. Module (B) is the e-hailing supply with Eq. (5.3). This module outputs the primary variable zωj (em ) along with its marginal prices λω (em ) into Modules (A) and (C). Within Module (A), the e-hailing supply and surge price affect e-hailing passengers waiting cost via the primary variable θ jω by Eq. (5.11). Receiving the input of e-hailing vehicle flows from Module (B), Module (C) contains Eq. (6.5a) describing the route choice of e-hailing vehicles in the path flow space. Module (D) includes Eqs. (6.3), (4.8) to model the route choice behavior of personal vehicles in the link-node space. The equilibria of these two modules are the (generalized) user equilibrium, i.e., Wardrop equilibrium (Wardrop, 1952). Both Modules (C) and (D) output vehicular flows in a road network and their resultant travel costs, which further affect travelers’ mode selection defined in Module (A). Modules (A) and (B) represent demand-side and supply-side of the e-hailing market. The exchange of variables between Modules (A) and (B) describes a market clearance mechanism which is widely used in market equilibrium modeling (Ferris and Pang, 1997). Accordingly, the equilibrium between Modules (A) and (B) is the so-called economic equilibrium. Traffic congestion arising from Modules (C-D) impacts travelers’ modal choice behavior in Module (A). As a result, travelers’ mode choice defined in Module (A) affects traffic flows in a road network. The mutual interactions among all four modules determines the multi-modal new mobility equilibrium, which is reached when no variables can improve their values.

Fig. 5. Model structure.

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6.2. Discussion on solution existence The existence conditions of the equilibrium is not trivial to establish because the overall equilibrium model is quite different from the existing equilibrium models. This game involves shared constraints that couple multiple players’ decision variables, which makes it challenging to demonstrate solution existence. Overall the major challenges involved in this game are: 1. coupled side constraints: the mode choice module outputs the modal demands, which serve as the side constraints to the other three modules; 2. nonconvex: nonlinear cost functions c˜, C in x, bilinear coupling between primary variables z and θ or between x and o in Eqs. (6.3d), (5.11a); 3. lack of bounds on the derived variables: the waiting cost cw,Oω , the marginal prices λω (em ) , the minimum generalized cost π ω (m ) , ∀m = {s, r, r, em } price tuples are not bounded. A convex game with shared constraints that couple the players’ decision variables is a “Generalized Nash Game” (Harker, 1991; Facchinei and Pang, 2010; Pang and Scutari, 2011). For a generalized Nash game, the basic idea of showing solution existence is to remove the shared constraints from players’ optimization problems and impose it as side constraints to be satisfied by a Nash Equilibrium, with the associated multipliers taken as prices in the players’ objective functions (Pang and Scutari, 2011). However, the proposed equilibrium is a nonconvex generalized game. When the players’ optimization problem is nonconvex, the equilibrium of a nonconvex game with side constraints is referred as “quasi-Nash equilibrium” (Pang and Scutari, 2011). Pang and Scutari (2011) developed a sufficiency condition for the uniqueness of quasi-Nash equilibrium. However, the conclusions cannot be directly applied to the developed game, partly because of the network congestion modules (C-D). Di et al. (2017, 2018) proposed sufficient conditions to ensure existence and uniqueness of ridesharing user equilibrium (RUE). Ban et al. (2018) proposed a procedure to prove the solution existence of a similar game with only solo driving and e-hailing service being considered. The underlying principle is to convert the original complementarity conditions into a penalized quasi-variational inequality (VI). The game proposed in this paper involves both e-hailing and RS, which adds additional layers of complexity in solution existence demonstration. This is because the sufficient conditions for RUE existence and uniqueness may not ensure the existence of e-hailing platform and vehicle route choice games. 6.3. Discussion on solution uniqueness It is challenging to prove the uniqueness of an equilibrium and is even harder to find multiple equilibria. Non-uniqueness may exist in: (i) passengers’ modal choice, (ii) solo or RS drivers role choice, (iii) personal vehicle or e-hailing drivers’ route choice, or (iv) e-hailing platforms’ order assigning choice. The non-uniqueness of equilibria may lead to different performance measures in a new shared mobility system. This is a general and important issue for all the network problems with multiple equilibria, especially when the modeling framework is used to predict network congestion in presence of multiple traveling modes. There exist a handful of papers addressing non-unique equilibrium problems. One option is to characterize the equilibrium solution set (Di et al., 2013; Han et al., 2015; Di and Liu, 2016). However, this is usually challenging due to non-convexity of an equilibrium set. Another option is to adopt various risk-taking attitudes in equilibrium solution selection for policy-making (Ban et al., 2009; Di et al., 2014, 2016). In such a framework, a bi-level model needs to be developed wherein the upper level is to select a risk-averse, risk-neutral, or risk-prone equilibrium solution and the lower level is our proposed equilibrium framework. This requires to design and model a mathematical program with equilibrium constraints (MPEC) (Luo et al., 1996), which is challenging to solve and will be left for future research. 6.4. Discussion on solution algorithm With the complex structure of the game, solving a system of coupled nonlinear complementarity problems (NCP) is highly challenging. In this paper, instead, we use an optimization software, GAMS (General Algebraic Modeling System Rosenthal and Brooke, 2007), to solve the numerical problems in both small- and medium-size networks. PATH is an effective solver built in GAMS for NCPs (Ferris and Munson, 20 0 0). For large-scale problems, customized algorithms exploiting the inherent structure of the model are required to speed up computation. As the main focus of this paper is a first step towards developing a theoretic framework to model the co-existence of multiple shared mobility modes and offer insights into the contributions of each travel mode to traffic congestion, the technical details of solution existence, uniqueness, and solution alogrithm will be left for future research when network design problem is modeled. 7. Numerical example We propose the following measures to describe the performance of the studied transportation system:

Deadhead miles: DH ≡

 

k∈D ω∈ m∈M

zkoω (em ) lkoω ,

(7.1a)

X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

Total traveler-hour: T H ≡



qω τ ω ,

67

(7.1b)

ω ∈

Total vehicle-hour: V H ≡

  k∈D ω∈ m∈M

zkoω (em )tkoω +





 

qω ( em ) t ω +

ω∈ m∈M



 

qω (sr ) t ω ,

(7.1c)

ω∈ m∈M

Service vehicle-hour (SVH) Total vehicle flow: N ≡



 

qω (sr ) +

ω ∈

(7.1d)

ω∈ m∈Me

 ˜ ≡ Vehicle-demand ratio: N

q ω ( em ) ,

ω ∈ q

 ω (sr ) +  qω (em )  ω∈ω m∈Me . ω ∈ q

(7.1e)

DH measures the miles traveled by vacant e-hailing vehicles. TH measures the total hours spent on traveling by all the travelers. VH measures the total hours consumed by vehicles including service vehicle-hour, which is the total hours both occupied and vacant vehicles spend on road. N is the total number of vehicles (including personal vehicles and e-hailing ˜ is the total number of vehicles discounted by the total demand level. vehicles) needed to transport a fixed travel demand. N 7.1. Braess network We will illustrate how to compute RUE in the Braess network. The topology of the network is illustrated in Fig. 6. The link performance functions are the Bureau of Public Roads (BPR) function, i.e.,



ti j ( xi j ) =

ti0j



xi j 1+A CAPACIT Yi j

B 

,

(7.2)

where xij is the vehicular flow, ti0j is the free-flow travel time, A, B are coefficients, and CAPACITYij is the link capacity (i.e., maximum vehicular flow). The coefficients in the link cost functions are listed in Table 5. The parameter values are listed in Table 6. Running GAMS 24.4.6 on a DELL desktop with Intel i7 CPU 3.40GHz and 16GB RAM, the computation time is 0.03 s. The algorithm stops with a residual less than 10−10 .

Fig. 6. The Braess network. Table 5 Cost coefficients. Link

LEN

ti0j

A

B

CAPACITY

(1,2) (1,3) (2,1) (2,3) (2,4) (3,1) (3,2) (3,4) (4,3) (4,1) (4,2)

10 20 10 20 10 20 20 20 20 40 10

0.3 0.5 0.3 0.4 0.4 0.5 0.4 0.3 0.3 1.0 0.4

0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15

4 4 4 4 4 4 4 4 4 4 4

40 40 40 60 40 40 60 40 40 60 40

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X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78 Table 6 Parameter values. Notations

Values

q41 , q14

100, 100 × sym 2 2

Cap FO(ωsr )

β1(s) , β1(r ) , β1(r ) β2(s) , β2(r )

1.2, 0.4, 0.3 1.8, 1.8 0.1,0.1 1,1 2 0.3,0.3 0,0 0.4,0.4 0.5, 0.45

α1(e1 ) , α1(e2 ) α2(e1 ) , α2(e2 ) β1(e1 ) , β1(e2 ) β2(e1 ) , β2(e2 )

0.3,0.38 1,1 0.1,0.1 0.1,0.1

(r )

(r ) , Iinc Iinc (r ) I p(r ) , I p p β1(e1 ) , β1(e2 ) β2(e1 ) , β2(e2 ) β3(e1 ) , β3(e2 ) FO(ωe1 ) , FO(ωe2 )

Table 7 Variables associated with ODs. OD

Demand mode flow q

(1,4) (4,1)

(sr)

59.33 2.89

q

( r)

3.06 2.89

Cost

q ( e0 )

q ( e1 )

π

τ

34.23 34.23

3.39

21.44 20.27

0.90 0.80

Table 8 Variables associated with links. xdi j(s ) 1 (1,2) (1,3) (2,4) (3,4) (3,2) (2,3) (2,1) (3,1) (4,2) (4,3) (4,1)

d (r )

xdi j(r ) 4 56.27

1

xdi j(e )

xi j 4 3.06

1

4 3.06

1

xdi j(z ) 4

1

37.62 56.27

2.45 0.61 0.55

2.45 0.61

37.62

0.55 0.61

0.61

2.77

2.77

2.23 0.55 0.12

2.23 0.55 0.12

33.66 0.56 33.66 0.56

3.39 3.39

tij

oij

0.52 0.56 0.68 0.34 0.40 0.40 0.34 0.50 0.46 0.30 1.00

1.00

4

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

The variables associated with ODs are provided in Table 7 (zero elements will be skipped). The first column lists two OD pairs. The first row lists the demands for driving personal vehicles q(sr) , RS riders q(r) , and two e-hailing services q(e0 ) , q(e1 ) , respectively, and the associated cost including generalized travel cost π (i.e., node potentials) and minimum travel time τ . For OD pair (1,4), among 100 units of travel demands, 59.33 units drive their personal vehicles, 3.06 ride others’ vehicles, and 34.23 and 3.39 choose e-hailing services provided by two companies, respectively. The generalized travel cost is 21.44 and the minimum travel time is 0.9. For OD pair (4,1), among 40 units of travel demands, 2.89 units drive their personal vehicles and share rides with 2.89 riders, and 34.23 units use e-hailing services provided by the first company. The generalized travel cost is 20.27 and the minimum travel time is 0.8. The values of selected link variables are provided in Table 8. The first column lists eleven links. The first row lists the relevant variables, including three types of flows, link travel time, and occupancy ratio. Note that a personal vehicle has a capacity of two (i.e., one personal vehicle can carry at most two riders excluding the driver), the occupancy ratio of all the links with non-zero flows is one, i.e., one personal vehicle only carries one rider. We would like to explain the columns describing e-hailing vehicle OD (i.e, xdi j(e ) ) and DO flows (i.e., xdi j(z ) ). There are only 34.23 units of travel demands choosing e-hailing service for OD pair (4,1), but there are 37.61(= 37.05 + 0.56 ) units of e-hailing vehicles. This is because 3.39 units of e-hailing vehicles from e-hailing provider 2 move vacantly to node 1 to serve OD pair (1,4). We can assume that due to lower cost of e-hailing service provided by provider 1, vehicles from provider 2 have difficulty in finding customers to travel on OD pair (4,1) and have to search passengers at node 1. Table 9 describes vehicle flow zωj (em ) , profit Rωj (em ) , and surge price λ associated with e-hailing vacant flows.

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Table 9 Variables associated with e-hailing vacant vehicle flows. zωj (em )

em = 1 (1,4)

1 4

λ

Rωj (em )

em = 2 (4,1)

(1,4)

(4,1)

34.23 1.52

34.23 0.00

3.39 1.61

1 4

em = 1

em = 2

(1,4)

(4,1)

(1,4)

(4,1)

18.47 16.39

15.86 17.95

18.44 16.36

15.87 17.96

0.00

Fig. 7. Link flows (sym = .4).

We visualize flows in Fig. 7. Fig. 7 a shows the solo driver flow. Note that the links which do not have solo drivers (i.e., links (1,3),(3,4)) are not plotted. Fig. 7b combines the flow distributions for ridesharing vehicles and passengers. The first number is the ridesharing driver flow while the one following in the parenthesis is the passenger flow. Fig. 7c plots the e-hailing vehicle flow by paths. The number 34.23(3.39) next to the demand arrow at node 4 indicates the number of e-hailing drivers originating from this node. The first number is the occupied driver flow while the one following in the parenthesis is the vacant flow. We would like to see the effects of three major cost related parameters on travelers’ mode choices prior to a trip. They are: 1. β2(sr ) : driving distance-based cost rate, affecting the cost of driving personal vehicles per unit of distance (e.g., fuel price). The higher, the fewer people who choose driving personal vehicles; 2. β2(e ) : e-hailing distance-based fare rate, affecting the cost of taking the e-hailing service per unit of distance. The higher, the fewer people who choose the e-hailing service; 3. p: RS link-based fee per unit of occupancy. The higher, the fewer drivers who choose to share a ride and the fewer riders who choose to ride a personal vehicle. For each of the above three parameter, we pick one low and one high levels of values and fix other two parameters as fixed. There are four combinations of three parameters, i.e., four scenarios in total shown in Table 10. In each scenario, we would also like to see how the demand symmetry coefficient sym affects the system performance. Define sym = d41 /d14 . The demand for OD pair (1,4) is fixed at 100 units, i.e., d14 = 100. Then d41 = sym · d14 = 100 · sym. When sym varies from 0 to 1, it means that the demand for OD pair (4,1) increases from 0 to 100.

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X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78 Table 10 Five types of entities. Scenario

β2(sr )

β2(e)

p

Description

0 1 2 3

1.8 1.8 1.8 1.5

1.0 0.5 1.0 1.0

2.0 2.0 0.0 2.0

benchmark low e-hailing fare low RS fee low cost of driving personal vehicles

Fig. 8a-b illustrate the pre-trip mode choice distribution for each scenario for OD pairs (1,4) and (4,1), respectively. Let us look at Scenario 0 in Fig. 8a. The x–axis is the demand symmetric coefficient sym varying from 0 to 1. The y–axis is the OD flow proportion among three mode choices: taking an e-hailing service, driving personal vehicles, or becoming a RS rider (i.e., riding a personal vehicle). A stacked bar represents the mode split proportion at a given sym value. The sum of three mode proportions is always one. For OD pair (1,4), a higher proportion of travelers choose e-hailing service, RS, or driving personal vehicles given a specific sym in Scenarios 1,2,3, respectively, compared to Scenario 0. The same trends appear for OD pair (4,1) as well. Another observation is that given a fixed sym < 1, the proportion of RS for OD pair (4,1) tends to be higher than that for OD pair (1,4). However, such discrepancy gradually shrinks as sym grows. For both OD pairs, as sym increases, more travelers choose e-hailing and fewer choose either to drive personal vehicles or being RS riders in Scenarios 0,1,2. This happens for OD pair (4,1) because of the increase in the total travel demand. When the total demand becomes higher, driving more personal vehicles can cause higher congestion cost. Thus a proportion of travelers start taking an e-hailing service. The reason for the growing demands of the e-haling service for OD pair (1,4) is not that obvious. When sym = 0 (i.e., there are 100 units of travel demands from origin 1 to destination 4 but no reverse demand), the e-hailing vehicles have to travel vacantly from destination 4 all the way back to origin 1 to get the next orders. It incurs high cost and low profit. Thus no e-hailing vehicle is willing to serve in the system. As sym gradually grows (i.e., more demands from origin 4 to destination 1), some e-hailing vehicles can continue to pick up orders at node 4 after their previous order ending at node 4, reducing the total operational cost and thus increasing the total profit. Thus the demands for an e-hailing service also grows. When sym = 1 (i.e., the system is completely symmetric in terms of travel demands), the mode split in all four scenarios for OD pair (1,4) is exactly the same as those for OD pair (4,1). In Scenario 3 (i.e., low cost of driving personal vehicles), nobody chooses e-hailing and the mode split between driving and RS is constant regardless of sym. Fig. 8c plots the generalized travel cost π for each scenario for each OD pair. The ranking of four scenarios in terms of generalized travel cost is preserved for both OD pairs. Travers share similar travel cost in both Scenarios 0 and 2 (i.e., low RS fee), which are the highest. The lowest travel cost is experienced in Scenario 1 (i.e., low e-hailing fare) when the majority of travel demands are fulfilled by e-hailing vehicles. The low fare of e-hailing service makes it a very attractive travel mode and thus a high proportion of people choose it. Accordingly the usage of such service reduces the generalized travel cost for each individual. Therefore Scenario 1 has the lowest generalized travel cost. As sym increases, the generalized travel cost decreases for OD pair (1,4) (because of the increasing demands for e-hailing service) but increases for OD pair (4,1) (due to higher demands). Fig. 9a-d depict the vacant e-hailing vehicle flow and their net profit at destination nodes 1,4 to serve two OD pairs, respectively, as sym increases. The figures in the left panel are the vacant e-hailing vehicle flow, net profit, and surge price in Scenario 0 (from top to bottom) and those in the right panel are for Scenario 1. Scenarios 2 and 3 are skipped to save space. We will mainly explain the figure in Scenario 1 (i.e., low e-hailing fare). In the first row, the two subfigures illustrate the vacant e-hailing vehicle flows originating from nodes 1 and 4, respectively, to serve OD pair (1,4). The x–axis is sym and the y–axis is the vacant vehicle flow. In the second row, the two subfigures illustrate the vacant e-hailing vehicle flows originating from nodes 1 and 4, respectively, to serve OD pair (4,1). As illustrated before, when sym is small, no e-hailing vehicle is available due to high return cost. As sym increases, there are more and more e-hailing vehicles available. Note that the vehicle flow from two e-hailing companies are complementary to each other. In other words, at a fixed sym, only one e-hailing company provides service for all the e-hailing passengers. This is partly because of the close fare and cost structure of two companies. The solver simply provides one set of solutions regarding the optimal e-hailing vehicle flows. This is partially confirmed by the surge price of two e-hailing services in Fig. 9f. In Scenario 0, these two services surge prices are almost the same. The uniqueness of solutions is quite challenging to guarantee and will be left for future research. The net profit of e-hailing service provider 1 is higher in general in Scenario 0 and that of provider 2 is higher in Scenario 1. High profit does not indicate high demand. It could discourage people from using the service. This is confirmed by the net profit per ride shown in Fig. 9c-d). In Scenario 1, provider 2 does not serve any demands even if the drivers earn higher profit. The higher profit is caused by setting higher fare, which discourages people to take service from provider 2. Thus there is no business for provider 2 at all in a competitive market. None of the scenarios has any vehicles which end up at node 1 but serve OD pair (4,1), partly because there are lower demands of OD pair (4,1) than that of OD pair (1,4). All the e-hailing vehicles who end up at node 1 have to carry passengers originating from node 1 and move to destination node 4 in the occupied state.

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71

Fig. 8. Numerical results.

Fig. 10e-a) illustrate the performance measures including the total number of vehicles needed to serve all the travel demands (according to Eq. (7.1d)) and the vehicle-demand ratio (according to Eq. (7.1e)), total vehicle-hour, service vehiclehour, and person-hour (according to Eq. (7.1b)–(7.1c)), deadhead miles traveled by vacant e-hailing vehicles (according to Eq. (7.1a)), as sym varies. A direct observation is that, the symmetric pattern of travel demand heavily influences DH: The more symmetric the total demand is, the less DH e-hailing incurs. In other words, a city with highly symmetric land use patterns could help reduce DH incurred by e-hailing vehicles. Scenario 1 has the highest DH because of its highest proportion of taking the e-hailing service. As sym increases to 1, DH is reduced to zero. Scenario 1 has the highest TH while Scenario 3 has the lowest SVH, which is consistent with minimum travel time shown in Fig. (8d). The emergence of e-hailing service lengthens individuals’ overall travel time. Among three

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X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

Fig. 9. Vacant vehicle flows.

modes, only RS can reduce the total number of required vehicles. Scenario 1 has the least RS travelers and thus the highest vehicle number. As sym increases, the total vehicle number goes up with the increase in travel demands. The same pattern is observed in the vehicle-demand ratio, because of the lowest RS flows in Scenario 1. In other words, when e-hailing service is relatively cheap, travelers tend to abandon driving and RS. When e-hailing only allows one passenger per trip, it can increase the total number of vehicles in the system. Accordingly, Scenario 1 has the highest SVH, which is obvious because of its highest number of e-hailing vehicles. When there are only service vehicles, SVH equals VH. Therefore SVH and VH for Scenario 1 coincide. By contrast, Scenario 1 has the lowest VH when sym is lower but a higher VH when sym is higher. For Scenarios 0 and 2, SVH increases monotonically due to the gradual increase in e-hailing vehicles, while VH first decreases and then increases as sym grows because of the increasing proportion of e-hailing vehicles. Scenario 3

X. Di and X.J. Ban / Transportation Research Part B 129 (2019) 50–78

Fig. 10. Numerical results (The legends of Scenarios 0,1,2,3 are listed from top to bottom).

73

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has almost zero SVH because it has no e-hailing vehicles, but has the highest VH due to the highest number of personal vehicles. Despite that the impact of the threshold of e-hailing vehicles on network performance cannot be derived analytically from the model, instead, mode choice proportion is endogenously determined by the model, we still would like to provide some insights into how the increasing penetration rate of e-hailing service may impact the overall system performance. Combining Figs. 8, 10, we find that the impact of e-hailing service on system performance is not monotone. In Scenario 0, 3.5% of total demand uses e-hailing when sym = 0.1 (see Fig. 8a) but not for sym = 0, 0.2, while the demand for ehailing starts to emerge and increase steadily from sym ≥ 0.3. At sym = 0.1, e-hailing service reduces generalized travel cost slightly (Fig. 8c) but increases minimum travel time (Fig. 8d), increases total vehicle-hour (Fig. 10d), and decreases vehicledemand ratio (Fig. 10f). When sym ≥ 0.3 as the proportion of using e-hailing service gradually increases, the generalized travel cost decreaeses, the minimum travel time increases, total vehicle-hour significantly decreases, and vehicle-demand ratio increases. In other words, when total demand level is relatively low (i.e., sym = 0.1), e-hailing service increases total vehicle-hour but decreases the total number of vehicles needed to serve one unit of demand (i.e., vehicle-demand ratio), while this impact is reversed if total demand level becomes higher (i.e., sym ≥ 0.3): a higher mode choice on e-hailing, leading to, however, lower vehicle-hour but higher vehicle-demand ratio. We would like to pinpoint that the conclusions obtained from this example may not be simply generalized to the realworld situations. For example, the high vehicle number in Scenario 1 may be attributable to the restricted assumption of one e-haling passenger per ride, i.e., no pooling is allowed in e-haling. If this assumption is relaxed, the number of total e-hailing vehicles may be greatly reduced, as argued for UberPool or LyftLine. Nevertheless, the conclusions can still provide insights into TNC regulation. Presently, e-hailing service cost less than taxi and sometimes even less than public transit, which attract travelers from these existing travel modes. Travelers who use e-hailing save travel cost. But the growing e-hailing vehicles also increase deadhead miles and traffic congestion, causing adverse environmental impacts. Therefore the emergence of e-hailing vehicles is a double-sided sword. On one hand, its low fare can lower people’s travel cost. On the other hand, it increases the total number of vehicles on road to serve the same level of travel demands. Therefore a lower bound on e-hailing price and an upper cap on driver quota need to be imposed. The low e-hailing price is partly caused by lack of regulation. One New York City taxi medallions cost over $1 million in 2013 and still $20 0,0 0 0 in 2018 (Perry, 2015). A TLC plate in NYC (Uber, 2018) cost less than $1,0 0 0. The entry into the e-hailing market is much easier, which enables lower e-haling fare. Lack of regulation benefits travelers for the time being, but can harm the system in the long run. In our example, the increasing number of e-hailing vehicles congests the road and thus makes everybody’s travel time longer. Moreover, the relative loose relation between drivers and e-hailing service providers may potentially cause mismatch between drivers and passengers and increase deadhead miles, which can bring adverse effects to the environment.

7.2. Sioux falls network The Sioux Falls network includes 24 nodes and 76 links. The topology of the network, node indexes, link indexes, and the link performance functions are downloadable from the github (https://github.com/bstabler/TransportationNetworks). We consider five origin nodes O = {1, 2, 4, 7, 9} and five destination nodes D = {13, 19, 20, 23, 24}, i.e., 25 OD pairs in total. The demands of all OD pairs are the same as 500. The values of the cost parameters for driving personal vehicles and RS remain the same as in the Braess network, except the value-of-time for solo drivers β1(s ) = 1.8. We increase it to ensure that all the traffic modes could be utilized in the system. Also, we only consider one e-hailing company and use the first set of cost parameters for e-hailing service. Then the subscript m in the parameters can be removed from now on. We will vary the time-based e-hailing fare rate α1 = {0, 0.1, 0.3, 0.5} for sensitivity analysis. The computation time of one equilibrium is on average 0.86 s. The algorithm stops with a residual less than 10−7 . Fig. 11a depicts passengers’ pre-trip mode choice distribution for each OD pair, as α 1 varies. Take the subfigure α1 = 0 for example. The x–axis is the OD index from 1 to 25 (as there are total 25 OD pairs). One bar represents the total demand for one OD pair. Different color indicates the portion of demands for each mode. Our first thought would be like: as α 1 increases, the proportion of e-hailing riders should decrease. However, that is not the case. Instead, the proportion of passengers who choose e-hailing first decreases, then increases and drops to zero as α 1 becomes higher. This happens due to a balance between demand and supply. The low e-hailing fare (i.e., α1 = 0) leads to a high demand for e-hailing service. On the other hand, low profit results in low supply, i.e., fewer e-hailing vehicles are available to passengers. This can be told from surge price. Fig. 11b demonstrates the surge price paid by e-hailing passengers for each OD pair for each α 1 . The x–axis is still the OD index, while for one OD pair there are four bars representing four scenarios, among which one bar represents the surge price of that OD pair for one scenarios. At α1 = 0, the gap between high demands and low supplies incurs high surge prices, which further increases passengers’ waiting cost and reduces the e-hailing demand. Coming back to Fig. 11a, we find that as α 1 gradually increases to 0.1, the number of e-haling passengers decreases due to higher fare. This number goes up again when α1 = 0.3, which is due to a well-balanced demand and supply of e-hailing service. This is confirmed by the lowest surge price shown in Fig. 11b. When α1 = 0.5, nobody is willing to take the expensive e-hailing service. Instead, solo vehicle flows and RS flows increase (as in Fig. 11e).

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Fig. 11. Numerical results (coe f = 3).

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The same trend across four scenarios are also illustrated further by other subfigures. Fig. 11c illustrates how many vacant vehicles originating from one destination to serve a particular OD pair. Still take the subfigure α1 = 0 for example. The x–axis is the OD index and one bar represents the total number of e-haling vehicles serving that OD pair. Different color indicates where these e-hailing vehicles come from. There are five destinations and thus there are five sources to provide vacant vehicles. Fig. 11d depicts the total number of vacant e-haling vehicles originating from each destination. The x–axis is four scenarios as α 1 varies. Fig. 11e depicts the multi-modal link flow for each scenario. The x–axis is the link index (as there are 76 links), while the y–axis is the modal link flow illustrated by four lines. Each line is one mode. The above observations tell us that lower e-hailing cost may not necessarily boost the demand for this mode, as it also discourages e-hailing drivers. This is caused by the bi-lateral nature of the sharing economy platform. When α1 = 0, the e-haling is too cheap that only a few driver want to work for the platform. As the e-hailing cost increases as α1 = 0.1, more drivers would like to work but fewer passengers want to take it, because the decrease in passengers’ waiting cost cannot compensate the increase in the total fare. When it continues to increase to α1 = 0.3, a good balance between supply and demand is reached, since the high number of drivers significantly reduce passengers’ waiting cost with a relatively milder increase in the total fare. At α1 = 0.5 though there are more drivers willing to serve, passengers would not choose this mode due to its high fare. Thus e-hailing related flows are all zero. From the perspective of e-hailing platform operators, solving for an optimal pricing structure is crucial to their business in a competitive market. Fig. 12a–c) illustrate performance measures including DH/VH/TH/SVH, the total number of vehicles needed to serve all the travel demands, and the generalized travel cost, as α 1 varies. Similarly, system travel time (including vacant time),

Fig. 12. Performance measures.

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deadhead miles, and the total number of vehicles first decrease, increase, and then drop as α 1 becomes larger. The increase in these measures is mostly caused by a higher number of e-hailing vehicles. DH and SVH drop to zero when α1 = 0.5 because of no e-hailing demand. The generalized travel cost is the highest when α1 = 0.5, indicating that driving personal vehicles increase overall travel cost while the e-hailing service can reduce it. Nevertheless, the least number of vehicles are required to serve the same travel demand level for α1 = 0.5 when there is no e-hailing service (as in Fig. 12b). Note that the generalized travel cost is the highest when α1 = 0.5 This result is somewhat consistent with that found in the first numerical example: decreasing e-hailing may improve system performance. On the other hand, the emergence of e-hailing can reduce travelers generalized travel cost, as in all other scenarios except α1 = 0.5. These analysis could help city planners and regulators to develop optimal policies to regulate e-hailing fare for social welfare. 8. Conclusions and future research This paper proposes a mixed link-node and a path formulation to model the equilibrium of new mobility systems with three modes: driving solo, ridesharing, and e-hailing service. To capture the complex interactions of these travel modes and their impacts on traffic congestion, a super extended network is created with four copied subnetworks each of whom represents one type of traffic flow: driving personal vehicles, ridesharing riders, e-hailing passengers, and e-hailing vehicles. In the personal vehicle flow network, parallel links are created to represent solo and RS vehicles connecting one node pair, enabling role switching between solo and RS drivers and passenger pick-up or drop-off at intermediate nodes. In the e-hailing vehicle flow network, parallel links are created to prevent the early passenger drop-off before reaching the passenger’s destination. Unlike personal vehicle flows, e-haling flows can be in two states: occupied and vacant. The former simply choose routes while the latter have to first select which OD pair to serve and then choose routes. Accordingly an e-hailing operational model is developed to assign vacant flows to OD pairs before their route choice is modeled using path flow. The existence conditions of the developed equilibrium are challenging because of the complex coupling among four modules. The underlying principle of illustrating solution existence for a quasi-variational inequality is briefly discussed. The equilibrium model is tested in the small-sized Braess network and the medium-sized Sioux Falls network. Sensitivity analysis is performed on parameters associated with modal cost, including distance-based and time-based e-hailing service fare, ridesharing fees, and personal vehicles driving cost. In the Braess network, four scenarios are compared, which correspond to benchmark, low e-hailing fare, low ridesharing fee, low driving cost. The overall performance measures such as modal cost, system travel time, and deadhead miles are used to evaluate the impact of each cost coefficient. A major finding is that lower e-hailing fare allows all the travelers to shift to using e-hailing service. Despite lower generalized travel cost, the total system travel time and deadhead miles can go up significantly due to the rebalance of vacant e-hailing vehicles if travel demands are highly imbalanced. Therefore regulations on pricing and driver entry should be imposed to the TNC industry. The outcomes of this study help assist transportation planners in making policy and regulation decisions regarding shared mobility services. Despite its novel modeling framework, this paper can be generalized in the following directions: (1) The link-node formulation provides a flexible modeling scheme for personal vehicle drivers to pick up or drop off RS riders at intermediate nodes. However, such flexibility may enable multiple drop-offs for one rider, which lengthens riders’ waiting time and increases their matching cost for the next available driver. These costs are not considered in the proposed model. Embedding them requires to define an appropriate matching function for ridesharing and will be left for the future research. (2) Analytical studies on equilibrium solution existence, uniqueness, and algorithm development will be explored in a more rigorous way. 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